In this article we study optimal control problems for systems that are affine with respect to some of the control variables and nonlinear in relation to the others. We consider finitely many equality and inequality constraints on the initial and final values of the state. We investigate singular optimal solutions for this class of problems, for which we obtain second order necessary and sufficient conditions for weak optimality in integral form. We also derive Goh pointwise necessary optimality conditions. We show an example to illustrate the results.

A. V. Dmitruk, Quadratic order optimality conditions for extremals completely singular in part of controls, in Operations Research Proceedings, Selected Papers of the Annual International Conference of the German Operations Research Society, (2011), 341–346.
doi: 10.1007/978-3-642-20009-0_54.

L. Poggiolini and G. Stefani, Minimum time optimality of a partially singular arc: Second order conditions, in Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, Lecture Notes in Control and Inform. Sci., 366, Springer, Berlin, 2007,281–291.
doi: 10.1007/978-3-540-73890-9_22.

A. V. Dmitruk, Quadratic order optimality conditions for extremals completely singular in part of controls, in Operations Research Proceedings, Selected Papers of the Annual International Conference of the German Operations Research Society, (2011), 341–346.
doi: 10.1007/978-3-642-20009-0_54.

L. Poggiolini and G. Stefani, Minimum time optimality of a partially singular arc: Second order conditions, in Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, Lecture Notes in Control and Inform. Sci., 366, Springer, Berlin, 2007,281–291.
doi: 10.1007/978-3-540-73890-9_22.

Ugur G. Abdulla.
On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences.
Inverse Problems & Imaging,
2016, 10
(4)
: 869-898.
doi: 10.3934/ipi.2016025